Finite Group Databasebeta

← #10690811

Subgroups — #10690811

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4705C211yesmaximal, characteristic
2#74257C211yesmaximal, characteristic
3#69807C211yesmaximal, characteristic
4C2×#4465C211yesmaximal, characteristic, complemented
5C2×#4465C211yesmaximal, characteristic, complemented
6C3×#1295C211yesmaximal, characteristic, complemented
7#67630C211yesmaximal, characteristic, complemented
8#9125—31nomaximal, Sylow
9C2×C42×D3C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C2×C3×#97C2211yescharacteristic
12C2×C3×#97C2211yescharacteristic
13C3×#382C2211yescharacteristic
14C2×(C2×D3)⋊C8—21no—
15C2×(C2×D3)⋊C8—21no—
16C2×Dic3⋊C8—21no—
17#4987C411yescharacteristic
18#4987C411yescharacteristic
19#4984C411yescharacteristic
20#4984C411yescharacteristic
21C2×#803C2211yescharacteristic
22C2×#803C2211yescharacteristic
23#4705—22no—
24#4705—22no—
25#4668—21no—
26#4668—21no—
27#4653—21no—
28#4653—21no—
29#4550—21no—
30#4465—22no—
31#4465—22no—
32C2×#432—31no—
33#2851—31no—
34#1749—31no—
35C2×#330—31no—
36C2×#330—31no—
37#1295—31no—
38#1295D311yescharacteristic, complemented
39C22×C4×D3C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3C2×C411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C22×C3⋊C8—21no—
59C22×C3⋊C8—21no—
60C42×D3—21no—
61C42×D3—21no—
62C42×D3—21no—
63C42×D3—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×#97—21no—
69C3×#97—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×#95—21no—
73C3×#95—21no—
74C3⋊#95—21no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78(C2×D3)⋊C8—24no—
79(C2×D3)⋊C8—24no—
80Dic3⋊C8—24no—
81#803—21no—
82#803—21no—
83#803—21no—
84#803D412yes—
85C2×#125—31no—
86C22×C42—31no—
87#432—62no—
88#432—62no—
89C2×C4⋊C8—61no—
90C2×C22⋊C8—61no—
91C2×C22⋊C8—61no—
92C2×#97—31no—
93C2×#97—31no—
94C2×#97C2×D311yescharacteristic
95C2×#97C2×D311yescharacteristic
96#389—31no—
97#389—31no—
98#387—61no—
99#387—61no—
100#385—61no—
101#385—61no—
102#382C2×D311yescharacteristic
103#380—31no—
104#380—31no—
105#336—61no—
106#330—62no—
107#330—62no—
108C23×D3C22⋊C411yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3—21no—
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C2×C4×D3—22no—
120C2×C4×D3—21no—
121C2×C4×D3—21no—
122C2×C4×D3—21no—
123C2×C4×D3—21no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—22no—
127C2×C4×D3—41no—
128C2×C4×D3—41no—
129C2×C4×D3—41no—
130C2×C4×D3—41no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C2×C3×C8—41no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4—42no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—42no—
150C4⋊Dic3—42no—
151Dic3⋊C4—41no—
152Dic3⋊C4—41no—
153Dic3⋊C4—41no—
154Dic3⋊C4—41no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159C4×Dic3—21no—
160C4×Dic3—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—22no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C23×C4—31no—
168C23×C4—31no—
169C23×C4—31no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—61no—
173C22×C8—61no—
174#125—61no—
175#125—61no—
176#125—61no—
177#125—61no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4C4×D311yescharacteristic
185C2×C4⋊C4C4×D311yescharacteristic
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—31no—
189C2×C42—31no—
190C2×C42—31no—
191C2×C42—61no—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196C4⋊C8—64no—
197C22⋊C8—64no—
198C22⋊C8—64no—
199#97—21no—
200#97—21no—
201#97—21no—
202#97—32no—
203#97—61no—
204#97—61no—
205#97—61no—
206#97D1212yes—
207#95—21no—
208#95—21no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3C2×C22⋊C411yescharacteristic
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—21no—
220C22×D3#10412yes—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3#10112yes—
254C4×D3—42no—
255C4×D3—42no—
256C4×D3—42no—
257C4×D3—42no—
258C4×D3—42no—
259C4×D3—42no—
260C3×C8—42no—
261C3×C8—42no—
262C3⋊C8—42no—
263C3⋊C8—42no—
264C24—31no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—62no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—61no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—121no—
284C22×C4C2×C4×D311yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4D3×D411yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—121no—
296C2×C8—121no—
297C2×C8—121no—
298C2×C8—121no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—122no—
306C4⋊C4—122no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C4⋊C4—121no—
311C22⋊C4—122no—
312C22⋊C4—122no—
313C42—61no—
314C42—61no—
315C42—61no—
316C42—61no—
317C42—61no—
318C42—61no—
319C42C2×D1211yescharacteristic
320C42#23511yescharacteristic
321C42D3×D411yescharacteristic
322C42#23911yescharacteristic
323C22×C3#38611yescharacteristic
324C22×C3C2×#10111yescharacteristic
325C22×C3#41311yescharacteristic
326C22×C3C2×#10411yescharacteristic
327C22×C3#42111yescharacteristic
328C22×C3#42111yescharacteristic
329C22×C3#42111yescharacteristic
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—81no—
344C3×C4—81no—
345Dic3—22no—
346Dic3—41no—
347Dic3—41no—
348Dic3—81no—
349Dic3—81no—
350C23—32no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23D3×C22⋊C411yescharacteristic
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—32no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—122no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—121no—
390C2×C4—121no—
391C2×C4C4×D1211yescharacteristic
392C2×C4#88211yescharacteristic
393C2×C4#88411yescharacteristic
394C2×C4#89311yescharacteristic
395C2×C4#89411yescharacteristic
396C2×C4#89511yescharacteristic
397C8—42no—
398C8—42no—
399C8—122no—
400C8—122no—
401C2×C3#152811yescharacteristic
402C2×C3#153211yescharacteristic
403C2×C3#153511yescharacteristic
404C2×C3#153711yescharacteristic
405C2×C3#187111yescharacteristic
406C2×C3#189711yescharacteristic
407C2×C3#190111yescharacteristic
408D3—42no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22#466511yescharacteristic
417C22#474511yescharacteristic
418C22D3×#10111yescharacteristic
419C22D3×#10411yescharacteristic
420C22#482111yescharacteristic
421C22#482111yescharacteristic
422C22#482111yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—62no—
430C4—81no—
431C4—81no—
432C4—121no—
433C4—121no—
434C4—241no—
435C4—241no—
436C3#912511yescharacteristic, complemented, Sylow
437C2—122no—
438C2#6834611yescharacteristic
439C2#6834811yescharacteristic
440C2#6836311yescharacteristic
441C2#6836911yescharacteristic
442C2#6875511yescharacteristic
443C2#6875711yescharacteristic
444C2#6899311yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1295C2
socleC23×C3C2×C22⋊C4
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1295D3
hypocenterC3#9125
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#4465C2semidirect
5. C2×#4465C2semidirect
6. C3×#1295C2semidirect
7. #67630C2semidirect
38. #1295D3semidirect
436. C3#9125semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#47052
#742572
#698072
C2×#44652
C3×#12952
#676302
#91253